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# Coordinate Geometry Word Problems Study Guide

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Updated on Oct 3, 2011

## Introduction to Coordinate Geometry Word Problems

Inspiration is needed in geometry, just as much as in poetry.

—ALEKSANDR PUSHKIN (1799–1837)

This lesson will cover the coordinate plane and important topics related to the coordinate plane, such as quadrants, slope, distance, and midpoint.

### The Coordinate Plane

The coordinate plane is made up of two perpendicular number lines that intersect, or cross, at a point known as the origin. Every point in the coordinate plane has a location determined by its coordinates, (x,y), and the origin has coordinates (0,0). When these lines intersect, they create four sections called quadrants. The quadrants are numbered, usually with Roman numerals, starting with the upper right-hand quadrant.

#### Tip:

The coordinates of points determine in which quadrant they are located. The first value in the parentheses is the x-coordinate and determines the number of units to move right (if a positive number) or left (if a negative number). The second value in the parentheses is the y-coordinate and determines the number of units to move up (if a positive number) or down (if a negative number).

An example of the coordinate plane with the origin and quadrants labeled is shown below.

Each point in the plane has a location given by two values in parentheses. To graph a point in the plane, start at the origin. Look at the first value. Move to the right that number of spaces if the first number is positive or to the left if the first number is negative. From there, look at the second value in parentheses. If the number is positive, move up that many spaces or down that many spaces if the number is negative.

For example, to graph the point A(4,5), start at the origin and move four units to the right and five units up. To graph the point B(2,–3), start at the origin and move two spaces to the right and three units down. To graph the point C(–3,1), move three units to the left and one unit up from the origin. To graph the point D(–1,–2), count one unit to the left and two units down from the origin. Each of these points is graphed in the figure.

#### Tip:

The formulas in coordinate geometry often use the values (x1,y1) and (x2,y2). Use the subscripts with x and y to keep track of each point.

## Slope Formula

The slope of a line can be described as the steepness of the line. Lines that tilt up to the right have positive slope. Lines that tilt up to the left have negative slope. Horizontal lines have zero slope, and vertical lines have undefined slope. To find the slope of a line, first take any two points on the line. Then, to find the slope between the two points, use the formula

Substitute the coordinates of the points into the formula and evaluate.

#### Example

What is the slope between the points (4,2) and (–3,7)?

Read and understand the question. This question is looking for the slope between two points.

Make a plan. Use the formula

and substitute the coordinates of the two points. Use (4,2) as (x1,y1) and (–3,7) as (x2,y2).

Carry out the plan. The formula becomes

The slope is .

Check your answer. To check this result, substitute the values in the opposite order to make sure that the slope is the same. The formula is

This result is checking.

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